Spaces with Ricci curvature bounded below

微分几何是数学的一个分支，研究光滑空间的性质，称为流形，它在数学、物理和其他科学学科中有着广泛的应用。通常，那些测量流形形状的量称为曲率。研究一种特殊的曲率--Ricci曲率--是微分几何的中心课题之一，与广义相对论有着重要的联系。这个项目致力于研究Ricci曲率不能太低的空间的几何和拓扑，即这种曲率有一个下界。Ricci曲率下界的概念也可以推广到可能没有流形结构的奇异空间(例如，具有锥尖的空间)。研究这些奇异空间可以反过来促进对光滑流形上Ricci曲率的理解。这个项目的更广泛的影响包括通过研讨会和会议报告提高对奇异空间重要性的认识，以一种容易理解的方式将几何中的现代概念和思想带给学生，以及指导这些领域的本科生。在一个方向上，该项目将研究具有非负Ricci曲率的完备和非紧致流形的基本群。具体地说，这将研究基本群的结构(例如，有限生成和虚拟幂零)与等变渐近几何之间的关系。该项目还将致力于了解Ricci极限空间的几何，特别是具有大Hausdorff维度的奇异集的几何。最后，该项目将探索Ricci曲率有界的闭流形的基本群，包括它们在Gromov-Hausdorff拓扑下的稳定性和对群结构的一致控制。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Differential geometry is a branch of mathematics that studies the properties of smooth spaces, known as manifolds, and it has profound applications throughout mathematics, physics, and other scientific disciplines. In general, those quantities that measure the shape of the manifold are called curvature. The study of one particular kind of curvature -- Ricci curvature -- is one of central topics in differential geometry and has significant connections to general relativity. This project seeks to study the geometry and topology of spaces whose Ricci curvature can't be too low, i.e., where this curvature has a lower bound. The notion of Ricci curvature lower bounds can also be extended to singular spaces (e.g., spaces with cone tips) which may not have manifold structures. Investigating these singular spaces can in turn advance the understanding of Ricci curvature on smooth manifolds. The broader impacts of this project include raising awareness of the importance of singular spaces through seminar and conference presentations, bringing modern concepts and ideas in geometry to students in an accessible way, as well as mentoring undergraduates in these areas.In one direction, the project will study the fundamental groups of complete and non-compact manifolds with nonnegative Ricci curvature. Specifically, this will investigate the relation between the structure of fundamental groups (for example, finite generation and virtual nilpotency) and the equivariant asymptotic geometry. The project will also aim to understand the geometry of Ricci limit spaces, especially that of singular sets with large Hausdorff dimension. Lastly, the project will explore the fundamental groups of closed manifolds with Ricci curvature bounded below, including their stability under the Gromov-Hausdorff topology and uniform control on the group structure.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.